Set and its Types

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SET: A set is a collection of well defined objects which are called its elements. For example: A = {a, e, i, o, u}

TYPES OF SET:

  1. Empty / Null Set: A set which does not contain any element is called an empty set, or the null set or the void set and it is denoted by ∅ or {} and is read as ‘phi’.

  2. Singleton Set: A set which contains only one element is called a singleton set. For example: Let B = {x : x is a even prime number}. Here B is a singleton set because there is only one prime number which is even, i.e., 2.

  3. Finite Set: A set which contains a definite number of elements is called a finite set. Empty set is also called a finite set, since the number of elements in an empty set is finite, i.e., 0. For example: A = {4, 5, 6, 7}

  4. Infinite Set: The set whose elements cannot be listed, i.e., set containing never-ending elements is called an infinite set. For example: A = {4, 5, 6, 7, …}

  5. Equal Set: Two sets A and B are said to be equal if they contain the same elements. Every element of A is an element of B and every element of B is an element of A. For Example: A = {p, q, r, s} and B = {p, s, r, q}.  Therefore, A = B.

  6. Equivalent Set: Two sets A and B are said to be equivalent if their cardinal number is same, i.e., n(A) = n(B). The symbol for denoting an equivalent set is ‘↔’.

    For example:

    A = {1, 2, 3}, Here n(A) = 3

    B = {p, q, r}, Here n(B) = 3.

    Therefore, A ↔ B

Subset: A set of which all the elements are contained in another set. The subset relationship is denoted as A⊂B. For example: A = {1, 2, 4} and B = {1, 2, 3, 4, 5}, means A⊂B P = {1, 2, 4} and Q = {3, 4, 5, 6, 7} means P⊄Q

Question: Find the subsets of C = {x, y, z}. Using formula, 2n = 23 = 2 ×2 ×2 = 8 C = { }, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}

Proper and Improper Subset: An improper subset is a subset containing every element of the original set. A proper subset contains some but not all of the elements of the original set. For Example: A = {1, 2, 3, 4, 5, 6}. B = {1, 2, 4} This is a proper subset. C = {1} This is a proper subset. D = {1, 2, 3, 4, 5} This is an improper subset.